Questions tagged [uniform]

The uniform distribution describes a random variable that is equally likely to take any value in its sample space.

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Distribution function of $1/X$ when $X$ is uniform on $[-1,1]$

(from The Probability Tutoring Book, C. Ash, p. 157) Find the density of $Y$ if $Y = 1/X$ and $X$ is uniform on $[-1,1]$. The distribution function given in the answer key is $$ F(y) = \begin{...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
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Intuition behind generating discrete random variables from a uniform random number generator

The following is an exercise from Rice's Mathematical Statistics and Data Analysis: This problem shows one way to generate discrete random variables from a uniform random number generator. Suppose ...
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X is Uniform $[-\theta,\theta]$ what is the distribution of $Y=\frac{1}{x^{2}}$?

X is Uniform $[-\theta,\theta], \theta>0$ what is the distribution of $Y=\frac{1}{x^{2}}$ So I've been working on some transformation questions; however, most of them have been one to one. I am a ...
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377 views

What does it mean for the uniform prior? [closed]

I wonder about the meaning of uniform prior of an unknown parameter. Any argumentation with detail explanation would be much appreciated.
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PDF of cosine of a uniform random variable with additional shift

I need to calculate the PDF of a random variable, which is quite similar to what was asked here. However, I have to deal with a shifted cosine function. Thus, my random variable is defined as $$Y:=cos(...
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1answer
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Help with the posterior of a uniform distribution with a parameter that is uniformly distributed

Here is the question: My main issue is with the marginal distribution of θ, I know that the sampling distribution is 1/(θ^n), but what interval do we integrate on, it can't be [0, 1] because that ...
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What distribution does the mean of a random sample from a Uniform distribution follow?

For example, let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=1,\theta-1/2 < x < \theta +1/2$. Clearly, $X_i \sim U(\theta-1/2 , \theta +1/2)$. Some intuition would suggest that $\bar{X}...
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1answer
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Computing $\mathbb{E}(S_n)$ and $\mathbb{V}(S_n)$ for Bernoulli data with a uniform probability parameter?

Take $U \sim \text{U}(0,1)$ as an underlying probability and generate $X_1,X_2,...,X_n \sim \text{Bern}(U)$ independent Bernoulli trials with this probability. The number of successes in the sample ...
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How to estimate the PDF of the logarithm of a uniformly distributed random variable?

This is a question I have to solve and need help with. I know it's usual to give pointers and hints so the OP can follow from there. Thus, I'll appreciate all input that shows me the way to go. Let $...
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Why is Entropy maximised when the probability distribution is uniform?

I know that entropy is the measure of randomness of a process/variable and it can be defined as follows. for a random variable $X \in$ set $A$ :- $H(X)= \sum_{x_i \in A} -p(x_i) \log (p(x_i)) $. In ...
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Express the posterior distribution: $P(L|X_{1:N})$ using Baye's Rule in terms of the Uniform Distribution

$f(Z; A, B) = \frac{1}{B-A+1}$ if $A ≤ Z ≤ B$, 0 otherwise $(1)$ $P(L) = f(L; 1, M)$, (the prior) $(2)$ $P(X|L) = f(X; 1, L)$ (the likelihood of a single license plate number X) $(3)$ We further ...
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Does runif (R) ever return 0/1 [closed]

The title says it all. Can it happen that runif (with bounds 0 and 1) returns 0 or 1 in R?
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Jointly Complete Sufficient Statistics: Uniform(a, b)

Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that ...
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How to find confidence interval for Uniform([a,1])?

Let $ U_1, \dots, U_n $ be a random sample of uniform distribution over $ [a,1] $. Construct a confidence interval for $ a $ with $ 1-\alpha = 0.95 $. I managed to show that $ T = \min\{U_i\} $ is ...
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What is the marginal probability distribution of the surface uniform sphere

I need to complete the following problem Let $~(X_1, X_2)~$ be a continuous random vector, with uniform density on the unit sphere $~\{(x_1,~ x_2,~ x_3) ~∈~ \mathbb R^3 ~:~ x_1^2 + x_2^2 + x_3^2 = 1\}...
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Is the t-test applicable when populations are sampled from uniform distributions?

I have two samples both taken from uniform distributions. The two samples differ in size. My first question is can I use the t-test to test the hypothesis that the two distributions are the same. Now ...
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Getting an (approx.) uniform distribution as a sum of two normal distributions

I am preparing a dataset for a simulation that will be used to study the dynamics of a toy galaxy computationally. This galaxy will include both ordinary and dark matter. Most of the mass will be ...
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1answer
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probability distribution of a sum of random variables [closed]

Suppose we have a random variable $X$ $P[X=-1]=1/3$, $P[X=0]=1/3$ and $P[X=1]=1/3$ now let $Y=X^2$ we have $n$ independent realizations of $Y$ $(Y_1, Y_2,......, Y_n)$ what is the probability ...
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Using Hoeffding's inequality on sum of uniform variables

I have the following problem: $X_1,...,X_n$ are i.i.d. $\sim U(-3,5)$ continuous uniform variables in the support between -3 and 5. $S := X_1 + ... + X_n$. I need to use Hoeffding's inequality to ...
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Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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1answer
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Approximate covariance of a uniform closure

I am interested to find a formula for approximate covariance of a uniform closure U(0,1)/summation{U(0,1)} for n independent U(0,1). I could derive an approximate expression for the variance = 1/(3n^2)...
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Uniform random variable is greater by a constant from another uniform random variable

I am trying to formulate the following question. X and Y are IID , uniform r.v. with ~U(0,1) What is the probability of P( X-Y-0.5 > 0) = ? 0.5 is a constant here and can be different. I do ...
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389 views

Can the difference of random variables be uniform distributed? [duplicate]

Given two random variables X and Y with some distribution D, is it possible to choose a D such that Z = X - Y is uniform? Is there a standard distribution D that would satisfy this?
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Uniform distribution and ordered statistics

Let $X_1,....,X_{n-1}$ be $(n-1)$ random variables following a Uniform distribution. If we note $X_{(1)},..,X_{(n-1)}$ the associated ordered statistics, I would like to prove that : $$U_i=X_{(i+1)}-...
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Is there a continuous function that accepts a single uniform random variable and returns two independent uniform random variables?

I can define a function $f(X) = (Y_1,Y_2)$ that accepts a random variable $X$ with a uniform distribution on $[0,1]$, and returns two independent uniform random variables $Y_1,Y_2$. This function ...
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How to check the correctness of calculations with a gamma distribution?

I’m reading Ponomareva, Roman, and Date (2015) and trying to generate vector $P$ of the $2Ns + 3$ probability weights: $$P =\{\underbrace{p_1, p_2, \ldots, p_s, p_1, p_2, \ldots, p_s,p_1, p_2, \ldots, ...
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Claims and questions regarding $n$-ball distribution?

CONTEXT In my research, I am utilizing an $n$-ball distributions along with two related distributions. I'd like to make certain I have a firm handle on the way to describe my three distributions. I ...
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Consecutive differences of a uniform law

Let $N>0$ be the number of considered samples. We draw $x_1, \ldots, x_n$ from a uniform distribution over $[0;1]$. We compute $y_1, \ldots, y_{n-1}$ the differences of the sorted $(x_i)_i$. I'd ...
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Expected frequency differences when sampling from a uniform?

Imagine I put people into different groups based on a uniformly distributed random variable $y = f(x)$ (e.g. microseconds of their arrival to a website). After a while, I observe how many people are ...
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Beta rectangular distribution

I need to sample data from a beta rectangular distribution. As far as I know, this distribution is a mixture of the beta and uniform distribution. I am using Python and in particular SciPy. There is a ...
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1answer
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Test of uniform distribution using KS-test and chi square in R

I want to test if a given sample $x$ of $n = 500$ continuous observations is uniformly distributed on a given interval of $[a,b]$ ($a = min(x)$ and $b = max(x)$). Therefore I would like to compare the ...
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1answer
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Transformation of Uniform(0,1) random variable [closed]

Let $X \sim U[0,1]$. Find the pdf of $Y=4\sqrt{X}(1-\sqrt{X})$. I have been studying transformation of random variables and came across this exercise. Can anyone provide me a hint on how to ...
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1answer
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Sampling small set of indices from a very large set

I have a set containing 100 millions of indices. In each iteration, I choose $k$ sub-samples from this set, and once I select the $k$-samples, I increase or decrease the probability for the selected ...
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Request for Explanation: Deriving Probability Density Function of a Maximum Likelihood Estimator of a Uniform Distribution [duplicate]

I am reviewing some practice problems and have both a question and its solution but am struggling to understand them and am hoping someone can help me. I am struggling to follow the logic for ...
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Vector with elements from a uniform distribution, to be made unit

I have a two dimensional constant vector $\mathbf{A} = \left < 2,1 \right>$. Also, I have a vector $\mathbf{e} = \left < \epsilon_x, \epsilon_y\right >$. Both $\epsilon_x$ and $\epsilon_y$ ...
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1answer
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Probability that $2\times2$ matrix of random variables is invertible

Let $X_1, X_2, X_3, X_4$ be random variables, and let $A$ be the following matrix: $$ \left[\begin{matrix} X_1 & X_2\\ X_3 & X_4 \end{matrix}\right]. $$ Assume that $X_1, X_2, X_3, X_4$ are ...
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Distribution of empirical frequency

Suppose that, for a given $n \in \mathbb{N}$, I draw points $x_1,...,x_n$ uniformly in $[0,1]$ and independently from each other. What would be the distribution of the empirical frequency of points ...
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Hypothesis test for $\theta$ in a $uniform(0,\theta)$ distribution

Suppose $X_1, \ldots, X_n$ is a random sample drawn from a $uniform(0,\theta)$ distribution. We will test $H_o: \theta = 3$ and $H_a: \theta = 2$. Use the test statistic $X_{(n)}$ and reject $H_o$ if ...
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1answer
572 views

Finding complete sufficient statistic

Let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such. I know that $T=(X_{(1)}, X_{(n)} )$ is a ...
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Uniform distribution problems

I was confused about this problem Two emergency response units, unit A and unit B, patrol uniformly and independent a 10-km stretch of moad. An emergency incident occurs on the road and its position ...
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Jeffreys prior for continuous uniform distribution

A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by: $f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ ...
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How can I know if the Data I have is uniformly distributed?

I was doing some data collection and I face a problem similar to the one in the following example. Let assume, I have 10 Machines $i \in \{1,...,10\}$ each one started after a given time from start ...
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1answer
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What is the probability that you can form a triangle with these three line segments?

Two numbers are randomly selected between $(0,1)$, uniformly and independently distributed. What is the probability that the three resulting line segments, which are obtained by cutting the interval ...
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1answer
671 views

Asymtotic distribution of the MLE of a Uniform

A property of the Maximum Likelihood Estimator is, that it asymptotically follows a normal distribution if the solution is unique. In case of a continuous Uniform distribution the Maximum Likelihood ...
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Are differences between uniformly distributed numbers uniformly distributed?

We roll a 6-sided die a large number of times. Calculating the difference (absolute value) between a roll and its preceding roll, are the differences expected to be uniformly distributed? To ...
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Uniform vs Beta(1,1) prior

Is there any difference in applying a uniform prior or a Beta(1,1) prior for your Bayesian analysis ?In which conditions is one preferred over the other ?
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Sum of two continuous random variables

Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2). What is the probability of R1>1 given that R1 +R2<2? -- What I've tried: I know that $$ P(R1&...
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What is the benefit of using permutation tests?

When testing some null versus alternative hypotheses by a test statistic $U(X)$, where $X = \{ x_i, ..., x_n\}$, apply the permutation test with the set $G$ of permutations on $X$ and we have a new ...
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Getting variance of function of two uniform RVs [duplicate]

Have two independent RV's $X$ and $Y$ sampled uniformly from $[0,1]$ and $C = (X-Y)^2$. Want $V(C$). Rewrote as $V((X-Y)^2) = V(X^2) - 4V(X)V(Y) + V(Y^2)$ but that's too messy. Is it correct to write ...

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